Searching open source (possibly) to perform multivariate limit

Question

I'm searching same open source software to perform this kind of limit (without restricting and executing the limit to a variable):

$$ \lim_{(x, y)\to(0, 0)}\frac{x^3y}{x^6+y^2} $$ I've seen sage and maxima, but i don't know if they can help me... Then, if not this, can they perform double and triple integrals?

Answer 1

The limit does not exist since:

$\textbf{Case 1:}$ Let $(x,y) = (x, x^3)$ then we have: $$\lim_{(x, y)\to(0, 0)}\frac{x^3y}{x^6+y^2}=\lim_{(x, x^3)\to(0, 0)}\frac{x^6}{x^6+x^6}=\frac{1}{2}$$ $\textbf{Case 2:}$ Let $(x,y) = (x, 0)$ then we have: $$\lim_{(x, y)\to(0, 0)}\frac{x^3y}{x^6+y^2}=\lim_{(x, 0)\to(0, 0)}\frac{x^3 \cdot 0}{x^6}=0$$

Answer 2

Not an open-source solution, but check this: https://www.wolframalpha.com/input/?i=Limit[(x^3 y)/(x^6 + y^2), {x -> 0, y -> 0}]

Answer 3

Not open source, but instead of WolframAlpha, I prefer Mathematica Online, you can sign up for free: https://www.wolframcloud.com/

Once in, write down

Limit[(x^3 y)/(x^6 + y^2), {x, y} -> {0, 0}]

and the answer is

Indeterminate